Question

The hypotenuse of a right triangle is 13 inches. If the long leg is 7 inches longer than the short leg, find the lengths of the leg using the Pythagorean theorem

Answer 1

Answer:

The length of the short leg is 5 inches and the leg of the long leg is 12 inches

Step-by-step explanation:

Let

a -----> the long leg of a right triangle

b ----> the short leg of a right triangle

c ----> the hypotenuse of a right triangle

we know that

Applying the Pythagoras Theorem

$c^{2} =a^{2}+b^{2}$ -----> equation A

we have that

$a=b+7$ -----> equation B

$c=13\ in$ ----> equation C

Substitute equation B and equation C in equation A and solve for b

$13^{2} =(b+7)^{2}+b^{2}$

$169=b^2+14b+49+b^2$

$2b^2+14b+49-169=0$

$2b^2+14b-120=0$

Solve the quadratic equation by graphing

The solution is $b=5\ in$

see the attached figure

Find the value of a

$a=b+7$ ----> $a=5+7=12\ in$

therefore

The length of the short leg is 5 inches and the leg of the long leg is 12 inches

Answer 2

The lengths of the leg : 12 and 5

Further explanation

Quadratic function is a function that has the term x²

The quadratic function forms a parabolic curve

The general formula is

$\large {\boxed {\bold {f (x) = ax ^ 2 + bx + c}}}$

where a, b, and c are real numbers and a ≠ 0.

There are 3 ways to solve quadratic equations

a. factoring b. perfect squared c. quadratic formula

We use the factoring method

Quadratic equation x² + bx + c = 0 is equivalent to quadratic equation

${\displaystyle {x ^ 2 + (p + q) \times \: pq}}$

where p + q = b and px q = c

for a = 1, the result of factoring is (x + p) (x + q)

If at an angle there is an angle of 90 degrees, this triangle is said to be a right angle

There is a hypotenuse in this triangle which is the sum of the two sides

c² = a² + b²

where c = hypotenuse

This formula is known as the Pythagorean theorem which states that:

the hypotenuse or the longest side in a right triangle equal to the sum of the squares of the other sides.

• The hypotenuse of a right triangle is 13 inches

then c = 13

•  If the long leg is 7 inches longer than the short leg

then a = b + 7

Then the phytagorean equation becomes:

(b + 7)² + b² = 13²

49+14b+b²+b²=169

49+14b+2b²=169

2b²+14b-120=0

b²+7b-60=0

factoring

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

b=5 or b=-12

we use positive number ⇒5

So :

the long leg = b+7=5+7=12

the short leg = b = 5

Learn more

the triangle angle

brainly.com/question/1611320

Prove the converse of the Pythagorean Theorem

brainly.com/question/1597216

Use Pythagoras' theorem

brainly.com/question/10488367

Keywords :  Pythagorean Theorem , hypotenuse, the long leg