We will solve some Problems on two tangents to a circle from an external point.
1. If OX any OY are radii and PX and PY are tangents to the circle, assign a special name to the quadrilateral OXPY and justify your answer.
Solution:
OX = OY, are radii of a circle are equal.
PX = PY, as tangents to a circle from an external point are equal.
Therefore, OXPY is a kite.
2. ∆XYZ is right angled at Y. A circle with centre O has been inscribed in the triangle. If XY = 15 cm and YZ = 8 cm, find the radius of the circle.
Solution:
Using Pythagoras’ theorem, we get
XZ = \(\sqrt{XY^{2} + YZ^{2}}\) = \(\sqrt{225 + 64 }\) cm = \(\sqrt{289}\) cm = 17 cm.
We draw OP ⊥ XY, OQ ⊥ YZ and OR ⊥ XZ.
Therefore, OP = OQ = OR = r, where r is the radius of the circle.
PYQO is a square.
Therefore, PY = YQ = r.
Therefore, XP = 15 cm – r and QZ = 8 cm – r.
Now, tangents drawn to a circle from an external point are equal.
Therefore, XR = XP = 15 cm – r and RZ = QZ = 8 cm – r.
But XR + RZ = XZ
⟹ 15 cm – r + 8 cm – r = 17 cm
⟹ 23 cm – 2r = 17 cm
⟹ 2r = 23 cm – 17 cm
⟹ 2r = 6 cm
⟹ r = 3 cm.
From Problems on Two Tangents to a Circle from an External Point to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Aug 10, 24 04:59 PM
Aug 10, 24 02:25 AM
Aug 10, 24 01:59 AM
Aug 10, 24 01:57 AM
Aug 06, 24 02:12 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.