The two sides of a triangle are 17 cm and 28 cm long, and the lengths of the median drawn to the third side is equal to 19.5 cm. Find the distance from an endpoint of this median to the longest side.

Answer

To find the distance from an endpoint of the median to the longest side of the triangle, we will use Apollonius's theorem. Apollonius's theorem states that for any triangle with sides $a$ , $b$ , and $c$ , and a median $m_a$ drawn to side $a$ , the following relationship holds:

b^2 + c^2 = 2m_a^2 + 2\left(\frac{a}{2}\right)^2

Given:

$b = 17$ cm
$c = 28$ cm
$m_a = 19.5$ cm

We need to find the length of side $a$ . Plugging the given values into Apollonius's theorem:

17^2 + 28^2 = 2(19.5)^2 + 2\left(\frac{a}{2}\right)^2

First, calculate $17^2$ and $28^2$ :

17^2 = 289

28^2 = 784

Next, calculate $2(19.5)^2$ :

19.5^2 = 380.25

2 \times 380.25 = 760.5

Substitute these values into the equation:

289 + 784 = 760.5 + 2\left(\frac{a}{2}\right)^2

Simplify the left-hand side:

1073 = 760.5 + 2\left(\frac{a}{2}\right)^2

Subtract 760.5 from both sides:

1073 - 760.5 = 2\left(\frac{a}{2}\right)^2

312.5 = 2\left(\frac{a}{2}\right)^2

Divide both sides by 2:

156.25 = \left(\frac{a}{2}\right)^2

Take the square root of both sides:

\frac{a}{2} = \sqrt{156.25}

\frac{a}{2} = 12.5

Multiply both sides by 2 to find $a$ :

a = 25

Now, we need to determine the distance from an endpoint of the median to the longest side. Since the median divides the side $a$ into two equal segments, each segment is:

\frac{a}{2} = \frac{25}{2} = 12.5 \text{ cm}

Thus, the distance from an endpoint of the median to the longest side is:

\boxed{12.5}

Key Concept

The key math concepts tested in this question include:

Triangle Properties: Understanding the properties of triangles, including the relationship between the sides and medians.
Median of a Triangle: The median divides the triangle into two smaller triangles of equal area. The length of the median is calculated using a specific formula involving the lengths of the triangle's sides.
Distance Measurement: The task requires finding a specific distance (from an endpoint of the median to the longest side), which involves geometric reasoning and possibly the application of the Pythagorean theorem or congruence of triangles.
Geometric Relationships: Understanding how the median interacts with the sides of the triangle and how to apply relevant geometric theorems.

These concepts involve both calculation and visualization skills within the context of triangle geometry.

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