The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
5 cm and 12 cm
Step 1: Introduce a variable. Let the base of the right triangle be \(b\) cm.
We are told that the altitude (height) is 7 cm less than the base, so:
Altitude = \(b - 7\) cm.
The hypotenuse is given as 13 cm.
Step 2: Use the Pythagoras theorem. In a right triangle,
\[(\text{base})^2 + (\text{altitude})^2 = (\text{hypotenuse})^2\]
Substitute the given expressions:
\[b^2 + (b - 7)^2 = 13^2\]
Step 3: Expand and simplify.
First expand \((b - 7)^2\):
\[(b - 7)^2 = b^2 - 14b + 49\]
Substitute into the equation:
\[b^2 + b^2 - 14b + 49 = 169\]
Combine like terms on the left:
\[2b^2 - 14b + 49 = 169\]
Move 169 to the left side:
\[2b^2 - 14b + 49 - 169 = 0\]
\[2b^2 - 14b - 120 = 0\]
Step 4: Simplify the quadratic equation. Divide the entire equation by 2:
\[b^2 - 7b - 60 = 0\]
Step 5: Factorise the quadratic. We look for two numbers whose product is \(-60\) and sum is \(-7\). These numbers are \(-12\) and \(+5\) because:
\[-12 \times 5 = -60, \quad -12 + 5 = -7\]
So we can write:
\[b^2 - 7b - 60 = (b - 12)(b + 5) = 0\]
Step 6: Solve for \(b\).
From \(b - 12 = 0\), we get \(b = 12\).
From \(b + 5 = 0\), we get \(b = -5\).
But a side length cannot be negative, so \(b = -5\) is rejected.
Therefore, the base is: \(b = 12\) cm.
Step 7: Find the altitude.
Altitude = \(b - 7 = 12 - 7 = 5\) cm.
Conclusion: The other two sides of the right triangle are 5 cm (altitude) and 12 cm (base).