In an isosceles right-angled triangle, the median from the vertex of the right angle is 5 cm. Find the area of the triangle.

Consider the main properties of the median in an isosceles right triangle:

the median of a right-angled triangle is half the hypotenuse, that is, it divides it in half;
is the height and bisector of the triangle;
splits a triangle into two equal sizes.
Consider a triangle ABO
∆ ABO is a right-angled triangle, since AO is also a height, which means it is perpendicular to the AC. In the AВO triangle:

AO = ВO = 5 cm,

angle AOB = 90 degrees.

Consider the CBO triangle
∆CBO is a right-angled triangle, since AO is also a height, which means it is perpendicular to the AC. In the CBO triangle:

CO = ВO = 5 cm,

COB angle = 90 °

Determine the area ABC
We found out that AO = 5 cm and CO = 5 cm.Thus, AC = AO + CO = 10 cm.

Let’s calculate the area ∆ ABC by the formula:

S = 1/2 * AC * BO = 1/2 * 10 * 5 = 25 sq. Cm.

You can find the area ∆ABС in simply using the first property of the median of a right triangle:

BO = 1/2 * AC, which means AC = 2BO = 2 * 5 = 10 cm.

Hence S∆ABC = 1/2 * 10 * 5 = 25 sq. Cm.

Answer: 25 sq. Cm