Question

THE ALTITUDE OF A RIGHT TRIANGLE IS 7 CM LESS THAN ITS BASE.THE HYPOTENUSE IS 13 CM . FIND THE OTHER TWO SIDES.

Answer 1

Answer:

Base = 12 CM

Altitude =  5 CM

Hypotenuse = 13 CM

Step-by-step explanation:

Hypotenuse = 13 CM

Let Altitude = a

Base = b

Given

Altitude = 7 CM less than its Base

ie a=b-7

Apply Pythagorean Theorem with an actual right triangle.

Pythagoras' theorem states that: a² + b² = c².

(b-7)² + b² = 13²

13 x 13 = b² + 7² - 2b7 + b²

169 = b² + 49 - 14b + b²

169 = 2b² - 14b + 49

2b² - 14b + 49 - 169 = 0

2b² - 14b - 120 = 0

2(b² - 7b - 60) = 0

b² - 7b - 60 = 0/2

b² - 7b - 60 = 0

We factorize by splitting the middle term method.

(We need to find two numbers whose sum is -7 and product is -60 and the numbers are 5 and -12)

So the above equation will be

b² + 5b - 12b - 60 = 0

b(b+5) - 12(b+5)=0

(b-12) x (b+5) = 0

so

b-12=0 or b+5 = 0

b cant be negative , so b = 12

SO Base = b = 12 CM

Altitude = b-7 = 12-7 = 5 CM

Answer 2

Given,

• Altitude of right triangle is 7 cm less than its base.
• Hypotenuse is 13 cm.

To find,

• The other two sides.

Solution,

• Let x be the base of the triangle
• Then altitude will be (x-7)

We know that,

$\sf{Base^2+Altitude^2=Hypotenuse^2}$

So, by pythagoras theorem,

${x}^{2} + ( {x - 7)}^{2} = {13}^{2} \\ \\ ⟹2 {x}^{2} - 14x + 49 = 169 \\ \\ ⟹2 {x}^{2} - 14x + 49 - 169 = 0 \\ \\ ⟹2 {x}^{2} - 14x - 120 = 0 \\ \\ ⟹2( {x}^{2} - 7x - 60) = 0 \\ \\ ⟹ {x}^{2} - 7x - 60 = \frac{0}{2} \\ \\⟹ {x }^{2} - 7x - 60 = 0 \\ \\ ⟹ {x}^{2} - 12x + 5x - 60 = 0 \\ \\ ⟹x(x - 12) + 5(x - 12) = 0 \\ \\ ⟹(x - 12)(x + 5) = 0$

So, x = 12 or x = -5

Since,the side of a triangle cannot be negative,so the base of the triangle is 12 cm.

And the altitude will be (12-7) = 5 cm