Question7. The hypotenuse of a right triangle is 15 cm . If the 2 other sides differ by 3 cm , what are their lengths?
Studdy Solution
STEP 1
1. We are given a right triangle.
2. The hypotenuse is cm.
3. The two other sides differ by cm.
4. We need to find the lengths of these two sides.
STEP 2
1. Set up equations based on the Pythagorean theorem and the given difference between the sides.
2. Solve the system of equations to find the lengths of the two sides.
STEP 3
Let's denote the two legs of the triangle as and , where . According to the problem, the hypotenuse cm, and the difference between the two legs is cm. Therefore, we can write:
STEP 4
Use the Pythagorean theorem, which states:
Substitute into the equation:
STEP 5
Substitute into the Pythagorean theorem equation:
STEP 6
Expand the equation:
So the equation becomes:
STEP 7
Combine like terms:
STEP 8
Subtract from both sides to set the equation to zero:
STEP 9
Divide the entire equation by 2 to simplify:
STEP 10
Solve the quadratic equation using the quadratic formula:
where , , and .
STEP 11
Calculate the discriminant:
STEP 12
Calculate the roots:
STEP 13
Find the two possible values for :
Since must be positive, we have .
STEP 14
Find using :
The lengths of the two sides are cm and cm.
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