Q3. 聽What is the area of the triang... [FREE SOLUTION]

91影视

Algebra 1

Carter, Cuevas, Day, Holiday, Luchin

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 801 pages

ISBN: 9780078884801

Chapter 10: Q3. (page 664)

What is the area of the triangle given below?
A $\sqrt{2} + 10 \sqrt{5}$
B $17 + 5 \sqrt{10}$
C $12 \sqrt{2} + 8 \sqrt{5}$
D $8 .5 + 2 .5 \sqrt{10}$

Short Answer

Expert verified

The area of the given triangle is $8.5 + 2.5 \sqrt{10}$ square units.

Option $D$ is correct.

Step by step solution

Step 1. State the rule of addition of radicals.

Two radicals can be only added if the radicand of the terms are same. If the radicand of the terms are same, then add the radicals by adding their coefficients.

That is, $a \sqrt{x} + b \sqrt{x} + c \sqrt{x} = (a + b + c) \sqrt{x}$

Step 2. State the rule of multiplication of radicals.

Two radicals can be only multiplied if the index of the terms are same. If the index of the terms are same, then the coefficient of the radicals are multiplied together and the radicands are multiplied together under a common radical symbol.

That is, $(a \sqrt{x}) (b \sqrt{y}) = (a 脳 b) \sqrt{x 脳 y}$

Step 3. State the concept of area of triangle.

Area of a right angle triangle is half of the product of its perpendicular height and base.

Let the length of the perpendicular height be 鈥� $h$ 鈥� and the length of the base be 鈥� $b$ 鈥�,

Then the area of the triangle ( $A$ ) is given as:

$A r e a (A) = \frac{1}{2} 脳 h 脳 b 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� 鈥� (1)$

Step 4. Calculate the area.Observe the figure given below.

From the figure,

The height(h) of the triangle is $2 \sqrt{2} + \sqrt{5}$ units.

The base( $b$ ) of the triangle is $3 \sqrt{2} + \sqrt{5}$ units.

Substitute $2 \sqrt{2} + \sqrt{5}$ as $h$ and $3 \sqrt{2} + \sqrt{5}$ as $b$ in (1) to get the required area.

$\begin{matrix} A r e a (A) = \frac{1}{2} 脳 h 脳 b \\ = \frac{1}{2} 脳 (2 \sqrt{2} + \sqrt{5}) 脳 (3 \sqrt{2} + \sqrt{5}) \\ = \frac{1}{2} [2 \sqrt{2} (3 \sqrt{2} + \sqrt{5}) + \sqrt{5} (3 \sqrt{2} + \sqrt{5})] 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆赌� [鲍蝉颈苍驳鈥夆赌夆赌� (a + b) (c + d) = a (c + d) + b (c + d)] \\ = \frac{1}{2} [(2 \sqrt{2}) (3 \sqrt{2}) + (2 \sqrt{2}) (\sqrt{5}) + (\sqrt{5}) (3 \sqrt{2}) + (\sqrt{5}) (\sqrt{5})] 鈥夆赌夆赌夆赌夆赌夆赌夆赌� \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆€夆€夆€� [鲍蝉颈苍驳鈥夆赌� a (b + c) = a (b) + a (c)] 鈥夆赌夆赌夆赌夆赌夆赌� \\ = \frac{1}{2} [(2 脳 3) (\sqrt{2} 脳 \sqrt{2}) + (2) (\sqrt{2} 脳 \sqrt{5}) + (3) (\sqrt{5} 脳 \sqrt{2}) + {(\sqrt{5})}^{2}] \\ = \frac{1}{2} [(6) {(\sqrt{2})}^{2} + (2) (\sqrt{2 脳 5}) + (3) (\sqrt{5 脳 2}) + {(\sqrt{5})}^{2}] \\ = \frac{1}{2} [6 (2) + 2 (\sqrt{10}) + 3 (\sqrt{10}) + 5] 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€� \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆€夆€夆€� [鲍蝉颈苍驳鈥夆赌� {(\sqrt{x})}^{2} = x] \\ = \frac{1}{2} [12 + 2 \sqrt{10} + 3 \sqrt{10} + 5] \\ = \frac{1}{2} [12 + 5 + 2 \sqrt{10} + 3 \sqrt{10}] \\ = \frac{1}{2} [17 + (2 + 3) \sqrt{10}] \\ = \frac{1}{2} [17 + 5 (\sqrt{10})] \\ = \frac{1}{2} (17 + 5 \sqrt{10}) \\ = \frac{1}{2} (17) + \frac{1}{2} (5 \sqrt{10}) \\ = 8.5 + 2.5 \sqrt{10} \end{matrix}$

Therefore, the area of the given triangle is $8.5 + 2.5 \sqrt{10}$ square units.

Option $D$ is correct.

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91影视

What is the area of the triangle given below?
A $\sqrt{2} + 10 \sqrt{5}$
B $17 + 5 \sqrt{10}$
C $12 \sqrt{2} + 8 \sqrt{5}$
D $8 .5 + 2 .5 \sqrt{10}$

Short Answer

Step by step solution

Step 1. State the rule of addition of radicals.

Step 2. State the rule of multiplication of radicals.

Step 3. State the concept of area of triangle.

Step 4. Calculate the area.Observe the figure given below.

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91影视

What is the area of the triangle given below?A2+105B17+510C122+85D8.5+2.510

Short Answer

Step by step solution

Step 1. State the rule of addition of radicals.

Step 2. State the rule of multiplication of radicals.

Step 3. State the concept of area of triangle.

Step 4. Calculate the area.Observe the figure given below.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

What is the area of the triangle given below?
A $\sqrt{2} + 10 \sqrt{5}$
B $17 + 5 \sqrt{10}$
C $12 \sqrt{2} + 8 \sqrt{5}$
D $8 .5 + 2 .5 \sqrt{10}$