Here we will learn how to prove the basic proportionality theorem with diagram.
A line drawn parallel to one side of a triangle divides the other two sides proportionally.
Given: In ∆XYZ, P and Q are points on XY and XZ respectively, such that PQ ∥ YZ.
To prove: \(\frac{XP}{PY}\) = \(\frac{XQ}{QZ}\).
Proof:
Statement |
Reason |
1. In ∆XYZ and ∆XPQ, (i) ∠YXZ = ∠PXQ (ii) ∠XYZ = ∠XPQ |
1. (i) Common angle (ii) Corresponding angles |
2. ∆XYZ ∼ ∆XPQ |
2. AA criterion of similarity. |
3. \(\frac{XY}{XP}\) = \(\frac{XZ}{XQ}\) |
3. Corresponding sides of similar triangles are proportional. |
4. \(\frac{XY}{XP}\) – 1 = \(\frac{XZ}{XQ}\) – 1 ⟹ \(\frac{XY - XP}{XP}\) = \(\frac{XZ - XQ}{XQ}\) ⟹ \(\frac{PY}{XP}\) = \(\frac{QZ}{XQ}\) |
4. By subtracting 1 from both sides of statement 3. |
5. \(\frac{XP}{PY}\) = \(\frac{XQ}{QZ}\) |
5. Taking reciprocals of both sides in statement 4. |
Solved examples using basic proportionality theorem:
1. If in a ∆XYZ, P and Q are two points on XY and XZ respectively such that XP = 4 cm, PY = 3 cm, XQ = = 6 cm, QZ = 4.5 cm and ∠XPQ = 40° then find ∠XYZ.
Solution:
Here, \(\frac{XP}{PY}\) = \(\frac{4 cm}{3 cm}\) = \(\frac{4}{3}\), and
\(\frac{XQ}{QZ}\) = \(\frac{6 cm}{4.5 cm}\) = \(\frac{4}{3}\)
Therefore, \(\frac{XP}{PY}\) = \(\frac{XQ}{QZ}\)
⟹ PQ ∥ YZ
Therefore, ∠XYZ = ∠XPQ = 40°.
2. In the given figure, if XP = 6 cm, YP = 2 cm, XQ = 7.5 cm, find QZ.
Solution:
By basic proportionality theorem,
\(\frac{XP}{PY}\) = \(\frac{XQ}{QZ}\)
⟹ \(\frac{6 cm}{2 cm}\) = \(\frac{7.5 cm}{QZ}\)
⟹ QZ = \(\frac{7.5 cm × 2}{6}\)
⟹ QZ = 2.5 cm.
3. At a certain time of the day, a man, 6 feet tall, casts his shadow 8 feet long. Find the length of the shadow cast by a building 45 feet high, at the same time.
Solution:
Let the length of the shadow of the building be x.
As the source of light is the sun, XZ ∥ PQ and, hence ∆YXZ ∼ ∆YPQ.
Therefore, \(\frac{\textrm{Height of the Man}}{\textrm{Height of the Building}}\) = \(\frac{\textrm{Length of Shadow Cast by the Man}}{\textrm{Length of Shadow Cast by the Building}}\)
⟹ \(\frac{6 ft}{45 ft}\) = \(\frac{8 ft}{x}\)
⟹ x = 60 feet.
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