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Difficulty: MediumTriangle Congruence, Similarity, and Theorems

In triangle ABCABC, the side lengths are AB=AC=5AB = AC = 5 and BC=6BC = 6. Point DD is the midpoint of side BCBC, and point EE lies on side ABAB such that segment DEDE is perpendicular to side ABAB. What is the length of segment DEDE?

Answer: 2.4

Answer

The length of segment DEDE is 2.42.4 (or the equivalent fraction 12/512/5).
The correct answer is 2.42.4 (or 12/512/5). In the isosceles triangle ABCABC with AB=AC=5AB = AC = 5, the median ADAD to the base BCBC is also an altitude. Since DD is the midpoint of BCBC, we have BD=3BD = 3. Applying the Pythagorean theorem to right triangle ABDABD gives AD=52−32=4AD = \sqrt{5^2 - 3^2} = 4. Since DEDE is perpendicular to ABAB, triangle AEDAED is a right triangle that shares angle AA with right triangle ADBADB. Therefore, triangle AEDAED is similar to triangle ADBADB. The ratio of the opposite side to the hypotenuse in both triangles must be equal: DEBD=ADAB\frac{DE}{BD} = \frac{AD}{AB}, which gives DE3=45\frac{DE}{3} = \frac{4}{5}, or DE=2.4DE = 2.4.

Step-by-Step Solution

1
Determine the properties of the altitude ADAD in the isosceles triangle ABCABC.
ADAD is perpendicular to BCBC, and BD=3BD = 3.
In an isosceles triangle, the median to the base is also the altitude to the base. Since DD is the midpoint of BCBC, BD=BC2=62=3BD = \frac{BC}{2} = \frac{6}{2} = 3, and ∠ADB=90∘\angle ADB = 90^\circ.
2
Calculate the length of segment ADAD using the Pythagorean theorem in right triangle ABDABD.
AD=4AD = 4
Applying the Pythagorean theorem to right triangle ABDABD gives AD2+BD2=AB2AD^2 + BD^2 = AB^2. Substituting the known lengths yields AD2+32=52AD^2 + 3^2 = 5^2, which simplifies to AD2=25−9=16AD^2 = 25 - 9 = 16, so AD=4AD = 4.
3
Find the length of segment DEDE using triangle similarity.
DE=2.4DE = 2.4
Since segment DEDE is perpendicular to side ABAB, ∠AED=90∘\angle AED = 90^\circ. The right triangles AEDAED and ADBADB share the angle at AA, so they are similar by AA similarity (△AED∼△ADB\triangle AED \sim \triangle ADB). This allows us to set up the ratio of corresponding sides: DEBD=ADAB\frac{DE}{BD} = \frac{AD}{AB}. Substituting the values gives DE3=45\frac{DE}{3} = \frac{4}{5}, which results in DE=125=2.4DE = \frac{12}{5} = 2.4.

Key Concept

Properties of isosceles triangles, the Pythagorean theorem, and right triangle similarity theorems.

Alternative Method

Alternatively, the length of DEDE can be found using the area of right triangle ABDABD. The area of triangle ABDABD is 12×BD×AD=12×3×4=6\frac{1}{2} \times BD \times AD = \frac{1}{2} \times 3 \times 4 = 6. The area can also be expressed using the hypotenuse ABAB as the base and DEDE as the height: Area=12×AB×DE=12×5×DE\text{Area} = \frac{1}{2} \times AB \times DE = \frac{1}{2} \times 5 \times DE. Setting these equal gives 52DE=6\frac{5}{2} DE = 6, which yields DE=2.4DE = 2.4.
Estimated Time:1m 30s
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Topics

Triangle Congruence, Similarity, and Theorems

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