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PQRS is a trapezium in which PQ ∥ RS. T is the midpoint of QR. TU is drawn parallel to PQ which meets PS at U. Prove that 2TU = PQ + RS. Given: PQRS is a trapezium in which PQ ∥ RS. T is the midpoint of QR. TU ∥ PQ and TU meets PS at U. To prove: 2TU = PQ + RS. Construction
Continue reading "Midpoint Theorem on Trapezium  Converse of the Midpoint Theorem"
Here we will prove that any straight line drawn from the vertex of a triangle to the base is bisected by the straight line which joins the middle points of the other two sides of the triangle. Solution: Given: Q and R are the midpoints of the sides XY and XZ respectively of
Continue reading "Straight Line Drawn from the Vertex of a Triangle to the Base Diagram"
Here we will show that the three line segments which join the middle points of the sides of a triangle, divide it into four triangles which are congruent to one another. Solution: Given: In ∆PQR, L, M and N are the midpoints of QR, RP and PQ respectively. To prove ∆PMN ≅ LNM
Continue reading "Four Triangles which are Congruent to One Another  Prove with Diagram"
Here we will learn how to solve different types of midpoint theorem problem. In the adjoining figure, find (i) ∠QPR, (ii) PQ if ST = 2.1 cm. Solution: In ∆PQR, S and T are the midpoints of PR and QR respectively. Therefore, ST = \(\frac{1}{2}\)PQ and ST ∥ PQ.
Continue reading "Midpoint Theorem Problem  Midpoint Theorem  Converse of Midpoint"
The straight line drawn through the midpoint of one side of a triangle parallel to another bisects the third side. Given: In ∆PQR, S is the midpoint of PQ, and ST is drawn parallel to QR. To prove: ST bisects PR, i.e., PT = TR. Construction: Join SU where U is the midpoint
Continue reading "Converse of Midpoint Theorem  Proof of Converse of Midpoint Theorem"
The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it. Given: A triangle PQR in which S and T are the midpoint of PQ and PR respectively. To prove: ST ∥ QR and ST = 1/2QR Construction: Draw RU ∥ QP such that
Continue reading "Midpoint Theorem AAS & SAS Criterion of Congruency Prove with Diagram"
Here we will prove that in any quadrilateral the sum of the four sides exceeds the sum of the diagonals. Solution: Given: ABCD is a quadrilateral; AC and BD are its diagonals. To prove: (AB + BC + CD + DA) > (AC + BD). Proof: Statement 1. In ∆ADB, (DA + AB) > BD.
Continue reading "Sum of Four Sides of a Quadrilateral Exceeds the Sum of the Diagonals"
Here we will prove that in a triangle the sum of any two sides is greater than twice the median which bisects the remaining side. Solution: Given: In ∆XYZ, XP is the median that bisects YZ at P. To prove: (XY + XZ) > 2XP. Construction: Produce XP to Q such that XP = PQ.
Continue reading "Sum Of Any Two Sides Is Greater Than Twice The Median  ProofDiagram "
Here we will solve the problem on inequalities in triangle. Let XYZ be a triangle in which XM bisects ∠YXZ. Prove that XY is greater than YM. As XM bisects ∠YXZ, we have ∠YXZ = ∠MXZ ............ (i) Also, in ∆XMZ, ∠XMY > ∠MXZ, as an exterior angle of a triangle is always
Continue reading "Problem on Inequalities in Triangle  Solution with Diagram"
Here we will solve different types of problems on comparison of sides and angles in a triangle. 1. In ∆XYZ, ∠XYZ = 35° and ∠YXZ = 63°. Arrange the sides of the triangle in the descending order of their lengths. ∠XZY = 180°  (∠XYZ + ∠YXZ) = 180°  (35° + 63°) = 180°  98°
Continue reading "Comparison of Sides and Angles in a Triangle  Geometrical Property"
Here we will prove that of all the straight lines that can be drawn to a straight line from a given point outside it, the perpendicular is the shortest. Given: XY is a straight line and O is a point outside it. OP is perpendicular to XY and OZ is an oblique. To Prove: OP
Continue reading "Perpendicular is the Shortest Theorem  Inequalities in Triangle"
Here we will prove that the sum of any two sides of a triangle is greater than the third side. Given: XYZ is a triangle. To Prove: (XY + XZ) > YZ, (YZ + XZ) > XY and (XY + YZ) > XZ Construction: Produce YX to P such that XP = XZ. Join P and Z. Statement 1. ∠XZP = ∠XPZ.
Continue reading "The Sum of any Two Sides of a Triangle is Greater than the Third Side"
Here we will prove that if two sides of a triangle are unequal, the greater side has the greater angle opposite to it. Given: In ∆XYZ, XZ > XY To prove: ∠XYZ > ∠XZY. Construction: From XZ, cut off XP such that XP equals XY. Join Y and P. Proof: Statement 1. In ∆XYP, ∠XYP =
Continue reading "Greater Side has the Greater Angle Opposite to It  Triangle Inequalit"
Here we will prove that if two angles of a triangle are unequal, the greater angle has the greater side opposite to it. Given: In ∆XYZ, ∠XYZ > ∠XZY To Prove: XZ > XY Proof: Statement 1. Let us assume that XZ is not greater than XY. Then XZ must be either equal to or less
Continue reading "Greater Angle has the Greater Side Opposite to It  Prove with Diagram"
Here we will prove that the equal sides YX and ZX of an isosceles triangle XYZ are produced beyond the vertex X to the points P and Q such that XP is equal to XQ. QY and PZ are joined. Show that QY is equal to PZ. Solution: In ∆XYZ, XY = XZ. YX and XZ are produced to P and
Continue reading "Theorem on Isosceles Triangle  Proof Involving Isosceles Triangles"
Here we will prove that if two given points on the base of an isosceles triangle are equidistant from the extremities of the base, show that they are also equidistance from the vertex. Solution: Given: In the isosceles ∆XYZ, XY = XZ, M and N points on the base YZ such that
Continue reading "Points on the Base of an Isosceles Triangle  Prove with Diagram"
Here we will show that the straight lines joining the extremities of the base of an isosceles triangle to the midpoints of the opposite sides are equal. Solution: Given: In ∆XYZ, XY = XZ, M and N are the midpoints of XY and XZ respectively.
Continue reading "Lines Joining the Extremities of the Base of an Isosceles Triangle"
Here we will prove that ∆PQR and ∆SQR are two isosceles triangles drawn on the same base QR and on the same side of it. If P and S be joined, prove that each of the angles ∠QPR and ∠QSR will be divided by the line PS into two equal parts.
Continue reading "Problem on Two Isosceles Triangles on the Same Base  Proof  Diagram"
Here we will solve some numerical problems on the properties of isosceles triangles Find x° from the given figures. In ∆XYZ, XY = XZ. Therefore, ∠XYZ = ∠XZY = x°. Now, ∠YXZ + ∠XYZ + XZY = 180° ⟹ 84° + x° + x° = 180° ⟹ 2x° = 180°  84° ⟹ 2x° = 96°
Continue reading "Problems on Properties of Isosceles Triangles  Find x° and y° "
Here we will prove that if the three angles of a triangle are equal, it is an equilateral triangle. Given: In ∆XYZ, ∠YXZ = ∠XYZ = ∠XZY. To prove: XY = YZ = ZX. Proof: Statement 1. XY = ZX. 2. XY = YZ. 3. XY = YZ = ZX. (Proved) Reason 1. Sides opposite to equal angles ∠XZY
Continue reading "Three Angles of an Equilateral Triangle are Equal  Axis of Symmetry"
Here we will prove that the sides opposite to the equal angles of a triangle are equal. Given: In ∆ABC, ∠XYZ = ∠XZY. To prove: XY = XZ. Construction: Draw the bisector XM of ∠YXZ so that it meets YZ at M. Proof: Statement 1. In ∆XYM and ∆XZM, (i) ∠XYM = XZM (ii) ∠YXM = ∠ZXM
Continue reading "Sides Opposite to the Equal Angles of a Triangle are Equal  Diagram"
Here we will prove that the three angles of an equilateral triangle are equal. Given: PQR is an equilateral triangle. To prove: ∠QPR = ∠PQR = ∠ PRQ. Proof: Statement 1. ∠QPR = ∠PQR 2. ∠PQR = ∠ PRQ. 3. ∠QPR = ∠PQR = ∠ PRQ. (Proved). Reason 1. Angles opposite to equal sides QR
Continue reading "The Three Angles of an Equilateral Triangle are Equal  With Diagram"
Here we will prove if the equal sides of an isosceles triangle are produced, the exterior angles are equal. Given: In the isosceles triangle PQR, the equal sides PQ and PR are produced to S and T respectively. To prove: ∠RQS = ∠QRT. Proof: Statement 1. ∠PQR = ∠PRQ 2. ∠RQS
Continue reading "Equal Sides of an Isosceles Triangle are Produced, the Exterior Angles"
Here we will prove that in an isosceles triangle, the angles opposite to the equal sides are equal. Solution: Given: In the isosceles ∆XYZ, XY = XZ. To prove ∠XYZ = ∠XZY. Construction: Draw a line XM such that it bisects ∠YXZ and meets the side YZ at M. Proof: Statement
Continue reading "Angles Opposite to Equal Sides of an Isosceles Triangle are Equal"
Here we will prove some Application of congruency of triangles. PQRS is a rectangle and POQ an equilateral triangle. Prove that SRO is an isosceles triangle. Solution: Given: PQRS is a rectangle. POQ is an equilateral triangle to prove ∆SOR is an isosceles triangle. Proof:
Continue reading "Application of Congruency of Triangles  Isosceles triangle Proved"
Here we will prove that the bisectors of the angles of a triangle meet at a point. Solution: Given In ∆XYZ, XO and YO bisect ∠YXZ and ∠XYZ respectively. To prove: OZ bisects ∠XZY. Construction: Draw OA ⊥ YZ, OB ⊥ XZ and OC ⊥ XY. Proof: Statement 1. In ∆XOC and ∆XOB
Continue reading "Prove that the Bisectors of the Angles of a Triangle Meet at a Point"
Here we will prove that any point on the bisector of an angle is equidistant from the arms of that angle. Solution: Given OZ bisects ∠XOY and PM ⊥ XO and PN ⊥ OY. To prove PM = PN. Proof: Statement 1. In ∆OPM and ∆OPN, (i) ∠MOP = ∠NOP. (ii) ∠OMP = ∠ONP = 90°
Continue reading "Point on the Bisector of an Angle  Corresponding Parts of a Triangles"
Here we will prove that an altitude of an equilateral triangle is also a median. In a ∆PQR, PQ = PR. Prove that the altitude PS is also a medina. Solution: Given in ∆PQR, PQ = PR and PS ⊥ QR.To prove PS is a median, i.e., QS = SR Proof: Statement 1. In ∆PQS and ∆PRS,
Continue reading "Prove that an Altitude of an Equilateral Triangle is also a Median"
Here we will learn how to prove different types of problems on congruency of triangles. 1. PQR and XYZ are two triangles in which PQ = XY and ∠PRQ = 70, ∠PQR = 50°, ∠XYZ = 70°, and ∠YXZ = 60°. Prove that the two triangles are congruent. Solution: In a triangle, the sum of
Continue reading "Problems on Congruency of Triangles Prove Two Triangles are Congruent"
Here we will learn different criteria for congruency of triangles. I. SAS (SideAngleSide) Criterion: If two triangles have two sides of one equal to two sides of the other, each to each, and the angles included by those sides are equal then the triangles are congruent.
Continue reading "Criteria for Congruency  SAS AAS  SSS  RHS  CPCTC"
In worksheet on graph of linear relations in x, y we will plot different types of equation in the xy plane. 1. Draw the graph for each of the following: (i) x = 2 (ii) x = 3 (iii) y = 4 (iv) y + 1 = 0 (v) 2x + 3 = 0 2. For each of the following, take three ordered pair
Continue reading "Worksheet on Graph of Linear Relations in x, y  Draw the Graph "
Two triangles are said to be congruent if they are exactly alike in all respects. If one triangle is placed on the other, the two triangles will coincide exactly with each other, i.e., the vertices of the first triangle will coincide with those of the second. In a pair of
Continue reading "Congruency of Triangles  Definition of Congruent Triangles"
In worksheet on slope and yintercept we will get different types of equations. On understanding slope and yintercept of graphs of linear relations in x,y. 1. Determine the slope and yintercept of the line graph for each of the following relations in x, y.
Continue reading "Worksheet on Slope and Yintercept  Determine the Slope & Yintercept"
In worksheet on plotting points in the coordinate plane we will plot different types of coordinate points in the xy plane. 1. Plot the points in the coordinate plane. (i) (6, 2) (ii) (3, 7) (iii) (2.5, 4.5) (iv) (5, 3.5)
Continue reading "Worksheet on Plotting Points in the Coordinate Plane Coordinate Graph"
Here we will learn how to solve different types of problems on slope and yintercept. 1. (i) Determine the slope and yintercept of the line 4x + 7y + 5 = 0 Solution: Here, 4x + 7y + 5 = 0 ⟹ 7y = 4x – 5 ⟹ y = 4/7x  5/7
Continue reading "Problems on Slope and Yintercept  Determine the Slope & Yintercept"
Here we will learn how to solve different types of problems on plotting points in the xy plane. 1. Plot the points in the same figure. (i) (3, 1), (ii) (5, 0), (iii) (3, 4.5), (iv) (1, 6), (v) (2.5, 1.5) Solution: Draw two mutually perpendicular lines X’OX and Y’OY
Continue reading "Problems on Plotting Points in the xy Plane  Plot the Points "
Here we will learn how to draw the graph of a linear relation between x and y is a straight line. So, the graph of y = mx + c is a straight line. We know its slope is m and yintercept is c. By knowing the slope and yintercept for a line graph, the graph can be easily drawn
Continue reading "Drawing Graph of y = mx + c Using Slope and yintercept  Examples"
Here we will learn how to draw the graph of standard linear relations between x, y. Graph of x = 0 Some of the orders pairs of values of (x, y) satisfying x = 0 are (0, 1), (0, 2), (0, 1), etc. All the points corresponding to these ordered pairs are on the yaxis because
Continue reading "Graph of Standard Linear Relations Between x, y  Graph of y = x"
The graph of y = mx + c is a straight line joining the points (0, c) and c/m Let M = (c/m, 0) and N = (0, c) and ∠NMX = θ. Then, tan θ is called the slope of the line which is the graph of y = mx + c. Now, ON = c and OM = c/m. Therefore, in the rightangled ∆MON, tan θ =
Continue reading "Slope of the Graph of y = mx + c  What is the Graph of y=mxc?"
If the graph of y = mx + c cuts the yaxis at P then OP is the yintercept of the graph, where O is the origin. If OP is in the positive direction of the yaxis, the intercept is positive. But if OP is in the negative direction of the yaxis, the intercept is negative.
Continue reading "yintercept of the Graph of y = mx + c  How to Find yintercept?"
Here we will learn how to draw Coordinate geometry Graph. When two variables x, y are related, the value(s) of one variable depends on the value(s) of the other variable. Let x, y be two variables related by 9x  3y + 4 = 0. Then, y = 3x + \(\frac{4}{3}\).
Continue reading "Coordinate Geometry Graph  Graph of Linear Relations Between x, y"
If the coordinates (x, y) of a point are given, one can plot in the Cartesian xy plane by taking the following steps. Step I: Observe the signs of the coordinates and determine the quadrant in which the point should be plotted. Step II: Take a rectangular Cartesian frame of
Continue reading "Plotting a Point in Cartesian Plane  Determine the Quadrant "
The xaxis (XOX’) and yaxis (YOY’) divided the xy plane in four regions called quadrants. The region of the plane falling in the angle XOY is called the first quadrant. The region of the plane falling in the angle X’OY is called the second quadrant. The region of the plane
Continue reading "Quadrants and Convention for Signs of Coordinates  Four Quadrants"
Take two intersecting lines XOX’ and YOY” in a plane which cut at O and are perpendicular to each other. Let P be a point in the plane. Draw perpendiculars from P to the line XoX’ and YoY’. Let them be PL and PM. Measure PL and PM in the same scale in mm, cm or m, etc.
Continue reading "Rectangular Cartesian Coordinates of a Point  Signs of Coordinates"
In elementary plane geometry a point is described by given it a name, such as P, Q or R. But in coordinate geometry, a point is described by its position in the plane. The position of a point is given by an ordered pair (a, b) of real numbers.
Continue reading "Coordinates of a Point  Cartesian Coordinate System  Ordered Pair"
Let a and b be two real numbers. (a, b) is called a pair of real numbers a, b. But (a, b) is called an order pair if (a, b) is different from (b, a). In the ordered pair (a, b), a is called the first entry or first coordinate and b is called the second entry or second
Continue reading "Ordered Pair  First Coordinate  Second Entry or Second Coordinate"
If x stands for any of the real numbers from the set R then x is a variable over R. For example, if x is the over number in an oneday international cricket match of 50 overs then x is a variable over the set {1, 2, 3, 4, ...., 48, 49, 50}. Suppose, x is the side of a square
Continue reading "Independent Variables and Dependent Variables  Variable  Formula"
In order to express raw data in the form of grouped data we use classes (or class intervals) for the values of the variables. Depending upon the method of grouping data, class intervals can be divided into two categories. (i) Overlapping Class Intervals: If the values of a
Continue reading "Class Interval  Overlapping and Nonoverlapping Class Intervals"
We will discuss here how to use Tally marks. To count the number of times a value of the variable appears in a collection of data, we use tally mark ( / ). Thus tally mark represents frequency. Observe the tally marks and the corresponding frequencies:
Continue reading "Tally Marks  Tally Mark Represents Frequency  Use of Tally Marks"
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