Height of a Right Triangle is 7 cm Less Than Its Base
Question:
The height of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, form the quadratic equation to find the base of the triangle.
Solution
Let the base of the triangle be
\[ x \text{ cm} \]
Then the height is
\[ (x-7)\text{ cm} \]
Given hypotenuse
\[ 13\text{ cm} \]
Using Pythagoras theorem,
\[ (\text{Base})^2+(\text{Height})^2=(\text{Hypotenuse})^2 \]
\[ x^2+(x-7)^2=13^2 \]
\[ x^2+x^2-14x+49=169 \]
\[ 2x^2-14x+49=169 \]
\[ 2x^2-14x-120=0 \]
Dividing by 2,
\[ x^2-7x-60=0 \]
Required Quadratic Equation
\[ \boxed{x^2-7x-60=0} \]
Answer
If \(x\) denotes the base of the triangle, then the required quadratic equation is
\[ \boxed{x^2-7x-60=0} \]
This equation can be solved to find the base of the triangle.