Chapter 10: Q. 49 (page 824)

Use the results of Exercise 45 to find the areas of the quadrilaterals $P Q R S$ specified in Exercises 46–49.
$P Q = 7, Q R = 8, R S = 8, S P = 9, ∠ R = 60^{◦}$

Short Answer

Expert verified

The areas of the quadrilaterals $P Q R S$ is $12 \sqrt{5} + 16 \sqrt{3}$ .

Step by step solution

Step 1. Given Information

Using the results of Exercise 45 we have to find the areas of the quadrilaterals $P Q R S$

$P Q = 7, Q R = 8, R S = 8, S P = 9, ∠ R = 60^{◦}$

Step 2. The Construction of quadrilateral by the given measurement is

To find the area of quadrilateral we firstly finding the area of $∆ S P Q and ∆ Q R S$ then add them.

Step 3. Firstly finding the area of ∆QRS.

From the diagram we can see that $a = 8, b = 8, ∠ C = 60^{o}$

$Area of ∆ S P Q = \frac{1}{2} \times a \times b \times \sin C Area of ∆ Q R S = \frac{1}{2} \times 8 \times 8 \times \sin 60^{o} Area of ∆ Q R S = 4 \times 8 \times \frac{\sqrt{3}}{2} Area of ∆ Q R S = 2 \times 8 \times \sqrt{3} Area of ∆ Q R S = 16 \sqrt{3}$

Step 4. Now finding the area of ∆SPQ.

Before finding the area of $∆ S P Q$ we firstly find the value of diagonal.

$a^{2} = b^{2} + c^{2} - 2 b c \cos A where b = 8, c = 8, A = 60^{o} a^{2} = {(8)}^{2} + {(8)}^{2} - 2 \times 8 \times 8 \times \cos 60^{o} a^{2} = 64 + 64 - 2 \times 8 \times 8 \times \frac{1}{2} a^{2} = 64 + 64 - 64 a^{2} = 64 a = \pm 8$

Step 5. Now finding the area of ∆QRS using the formula Area of ∆QRS=s(s-a)(s-b)(s-c).

$s = \frac{a + b + c}{2} s = \frac{7 + 9 + 8}{2} s = \frac{24}{2} s = 12$

$Area of ∆ Q R S = \sqrt{12 (12 - 7) (12 - 8) (12 - 9)} Area of ∆ Q R S = \sqrt{4 \times 3 \times 5 \times 4 \times 3} Area of ∆ Q R S = 4 \times 3 \sqrt{5} Area of ∆ Q R S = 12 \sqrt{5}$

Step 6. Now finding the area of quadrilateral PQRS

$Area of P Q R S = Area of ∆ S P Q + Area of ∆ Q R S Area of P Q R S = 12 \sqrt{5} + 16 \sqrt{3}$

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Use the results of Exercise 45 to find the areas of the quadrilaterals $P Q R S$ specified in Exercises 46–49.
$P Q = 7, Q R = 8, R S = 8, S P = 9, ∠ R = 60^{◦}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The Construction of quadrilateral by the given measurement is

Step 3. Firstly finding the area of ∆QRS.

Step 4. Now finding the area of ∆SPQ.

Step 5. Now finding the area of ∆QRS using the formula Area of ∆QRS=s(s-a)(s-b)(s-c).

Step 6. Now finding the area of quadrilateral PQRS

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Use the results of Exercise 45 to find the areas of the quadrilaterals PQRSspecified in Exercises 46–49.PQ=7,QR=8,RS=8,SP=9,∠R=60◦

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The Construction of quadrilateral by the given measurement is

Step 3. Firstly finding the area of ∆QRS.

Step 4. Now finding the area of ∆SPQ.

Step 5. Now finding the area of ∆QRS using the formula Area of ∆QRS=s(s-a)(s-b)(s-c).

Step 6. Now finding the area of quadrilateral PQRS

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Applied Mathematics

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Use the results of Exercise 45 to find the areas of the quadrilaterals $P Q R S$ specified in Exercises 46–49.
$P Q = 7, Q R = 8, R S = 8, S P = 9, ∠ R = 60^{◦}$