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Question

The altitude of a right triangle is $$7$$ cm less than its base. If the hypotenuse is $$13$$ cm, find the other two sides.


Solution

Let the base be $$x$$ cm. Then the altitude will be $$x-7$$ cm.

By Pythagoras theroem,

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

Length cannot be negative, then the base is $$12 \ cm$$ and the height is $$5 \ cm$$

Mathematics
RS Agarwal
Standard X

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