ABC is a triangle right angled at B. Let D be the midpoint on AC. If BD = 6.5 cm, then what is AB 2 + BC 2 equal to?
169 square cm
Understanding the Right-Angled Triangle Problem
The question describes a triangle ABC that is right-angled at B. We are given that D is the midpoint of the hypotenuse AC, and the length of the median BD is 6.5 cm. We need to find the value of \(AB^2 + BC^2\).
Applying Key Geometric Properties
In a right-angled triangle, there are two fundamental properties that are crucial here:
- The Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. For triangle ABC right-angled at B, this means \(AB^2 + BC^2 = AC^2\).
- The property of the median to the hypotenuse: In a right-angled triangle, the median drawn from the right angle vertex to the hypotenuse is half the length of the hypotenuse. In triangle ABC, BD is the median to the hypotenuse AC. Therefore, \(BD = \frac{1}{2} AC\).
Step-by-Step Solution
We are given that BD = 6.5 cm.
Using the property of the median to the hypotenuse:
\(BD = \frac{1}{2} AC\)
Substituting the given value of BD:
\(6.5 = \frac{1}{2} AC\)
To find the length of AC, we multiply both sides by 2:
\(AC = 2 \times 6.5\)
\(AC = 13\) cm
Now we need to find \(AB^2 + BC^2\). According to the Pythagorean Theorem in the right-angled triangle ABC:
\(AB^2 + BC^2 = AC^2\)
We have found that AC = 13 cm. Substituting this value into the equation:
\(AB^2 + BC^2 = (13)^2\)
\(AB^2 + BC^2 = 13 \times 13\)
\(AB^2 + BC^2 = 169\)
The value of \(AB^2 + BC^2\) is 169 square cm.
Summarizing the Calculation
| Given Information | Property Used | Calculation | Result |
|---|---|---|---|
| Triangle ABC, right-angled at B | Median to hypotenuse BD = 6.5 cm | Median property: \(AC = 2 \times BD\) | \(AC = 2 \times 6.5 = 13\) cm |
| Triangle ABC, right-angled at B | Required: \(AB^2 + BC^2\) | Pythagorean Theorem: \(AB^2 + BC^2 = AC^2\) | \(AB^2 + BC^2 = (13)^2 = 169\) |
Therefore, \(AB^2 + BC^2\) is equal to 169 square cm.
Revision Table: Right Triangle Geometry
| Concept | Description | Formula/Property |
|---|---|---|
| Right-Angled Triangle | A triangle with one angle measuring 90 degrees. | - |
| Hypotenuse | The side opposite the right angle; it is the longest side. | - |
| Pythagorean Theorem | In a right triangle, the square of the hypotenuse length is the sum of the squares of the other two sides. | \(a^2 + b^2 = c^2\) (where c is hypotenuse) |
| Median to Hypotenuse | The line segment from the right angle vertex to the midpoint of the hypotenuse. | Length of median = \(\frac{1}{2}\) \(\times\) Length of hypotenuse |
| Midpoint | A point that divides a line segment into two equal parts. | - |
Additional Information: Converse Properties
It's also useful to know the converses of these properties:
- Converse of Pythagorean Theorem: If, in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
- Converse of Median to Hypotenuse Property: If the median to one side of a triangle is half the length of that side, then the triangle is a right-angled triangle, and that side is the hypotenuse.
These converses help in identifying right triangles based on side lengths or median lengths.
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