Q19. 鈯橮聽and 鈯橯聽have radii 5聽an... [FREE SOLUTION]

91影视

Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 9: Q19. (page 331)

$鈯� P$ and $鈯� Q$ have radii $5$ and $7$ and $P Q = 6$ . Find the length of the common chord $\overset{炉}{A B}$ . $A P B Q$ is a kite and $\overset{炉}{P Q}$ is he perpendicular bisector of $\overset{炉}{A B}$ . Let be the intersection of $\overset{炉}{P Q}$ and $\overset{炉}{A B}$ .Let $P N = x$ and $A N = y$ . Write two equations in term of x and y.

Short Answer

Expert verified

The length of the common chord $A B$ is $4 \sqrt{6}$ units.

Step by step solution

Step 1. Given information.

Two circles $P$ and $Q$ with radii $5$ and $7$ and $P Q = 6$ .

Step 2. Concept used.

Join $A P$ and $A Q$ , so that these are the radii of two circles. Further by property, any perpendicular from center to a chord of a circle is always perpendicular on it.

So, apply Pythagorean theorem and then we get two right triangles $螖 A N P$ and $螖 A N Q$ .

Step 3. Solution.

$\begin{matrix} L e t 鈥夆赌� P N = x, \\ A N = y, \\ N Q = 6 鈭� x \end{matrix}$

So, in right triangle $螖 A N P$ , using Pythagoras theorem,

$\begin{matrix} A P^{2} = A N^{2} + P N^{2} \\ 5^{2} = y^{2} + x^{2} \\ y^{2} + x^{2} = 25 鈥夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€� ...... (i) \end{matrix}$

Again in right triangle $螖 A N Q$ , using Pythagoras theorem,

role="math" localid="1648894640214" $\begin{matrix} A Q^{2} = A N^{2} + Q N^{2} \\ 7^{2} = y^{2} + {(6 - x)}^{2} \\ y^{2} + x^{2} 鈭� 12 x + 36 = 47 鈥夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€� ...... (i i) \end{matrix}$

By (i) and (ii),

$\begin{matrix} 25 鈭� 12 x + 36 = 49 \\ 61 鈭� 12 x = 49 \\ 鈭� 12 x = 49 鈭� 61 = 鈭� 12 \\ x = 1 \end{matrix}$

So, by equation (i),

$\begin{matrix} 1^{2} + y^{2} = 25 \\ y^{2} = 24 \\ y = A N \\ A N = \sqrt{24} \\ A N = 2 \sqrt{6} \end{matrix}$

$P Q$ bisects chord $A B$ .

Therefore,

$\begin{array}{l} A B = 2 A N \\ A B = 4 \sqrt{6.} \end{array}$

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91影视

鈯�Pand 鈯�Qhave radii 5and 7and PQ=6. Find the length of the common chord AB炉. APBQ is a kite and PQ炉is he perpendicular bisector of AB炉. Let be the intersection of PQ炉 and AB炉.Let PN=xand AN=y. Write two equations in term of x and y.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. Solution.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Applied Mathematics

Statistics

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Logic and Functions

Study anywhere. Anytime. Across all devices.