Chapter 9: Q14. (page 342)

a. Given: $\bar{W Z}$ is a diameter of $⊙ o; \bar{O X} ∥ \bar{Z Y}$
Prove: $\overset{⏜}{W X} ≅ \overset{⏜}{X Y} .$
(Hint: draw $\overset{⏜}{O Y} .$ )

Short Answer

Expert verified

(a)

The final answer is

Arc length $W X ≅ arc length XY.$

Step by step solution

Step 1. Given information:

$\bar{W Z}$ is a diameter of circle and $\bar{O X} ∥ \bar{Z Y}$

Step 2. Concept used:

We use basic geometric and angle concept

Step 3. Applying the concept:

First let us draw $\bar{O Y}$ in the given figure.

Then $\bar{O Y}$ and $\bar{O Z}$ becomes radii of the circles $\Rightarrow \bar{O Y} = \bar{O Z}$

Let $∠ W O X = x °$ , since $\bar{O X} | | \bar{Z Y}$ then $∠ O Z Y = x °$ and also $Δ O Z Y$ is isosceles triangles as $\bar{O Y} = \bar{O Z}$ then $∠ O Y Z = x °$

From $Δ O Z Y$ , we have $∠ O Z Y + ∠ Z Y O + ∠ Y O Z = 180^{0}$

$\begin{array}{l} \Rightarrow x^{\circ} + x^{\circ} + ∠ Z O Y = 180^{\circ} \\ \Rightarrow ∠ Z O Y = 180^{\circ} - 2 x^{\circ} \end{array}$

Also from straight line $\bar{W Z}$ , we have $∠ W O X + ∠ X O Y + ∠ Z O Y = 180^{0}$

$\begin{matrix} \Rightarrow x^{\circ} + ∠ X O Y + 180^{\circ} - 2 x^{\circ} = 180^{\circ} \\ \Rightarrow ∠ X O Y - x^{\circ} = 180 - 180 \\ ∴ ∠ X O Y = x^{\circ} \end{matrix}$

Thus, $∠ W O X = x ° = ∠ X O Y$

$\Rightarrow$ Arcs made by same angle in a circle are congruent.

Hence, Arc length $W X ≅$ arc length $X Y$ .

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a. Given: $\bar{W Z}$ is a diameter of $⊙ o; \bar{O X} ∥ \bar{Z Y}$
Prove: $\overset{⏜}{W X} ≅ \overset{⏜}{X Y} .$
(Hint: draw $\overset{⏜}{O Y} .$ )

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Concept used:

Step 3. Applying the concept:

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a. Given:WZ¯is a diameter of ⊙o;OX¯∥ZY¯ Prove: WX⏜≅XY⏜. (Hint: draw OY⏜.)

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Concept used:

Step 3. Applying the concept:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

a. Given: $\bar{W Z}$ is a diameter of $⊙ o; \bar{O X} ∥ \bar{Z Y}$
Prove: $\overset{⏜}{W X} ≅ \overset{⏜}{X Y} .$
(Hint: draw $\overset{⏜}{O Y} .$ )