## Friday, July 26, 2024

### Geometry Problem 1572: Calculate Angle C with Geometric Methods Using Sides AB=17, BC=25, and Angle A=45 Degrees

Challenging Geometry Puzzle: Problem 1572. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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1. the problem want us to use geometric methods rather than trigonometry to enhance understanding of the triangle's properties.

2. Draw a 7-24 -25 Right Triangle BCE with BE = 24, CE = 7 and BC = 25
So CE // AB and hence CDE & ABD are both right isosceles

From Problem 1571, < BCE = 74 and so
< C = 74 - 45 = 29

Sumith Peiris
Moratuwa
Sri Lanka

3. A better Pure Geometry proof

Draw BD perpendicular to AB, D on AC. Draw CE // AB, E on BD extended
Now both Triangles ABD & CDE are right isosceles

Let CE = DE = x

Using Pythagoras in Triangle BCE,
x^2 + (x + 17)^2 = 25^2
So we have a quadratic equation in x
2x^2 + 34x - 336 = 0, from which
x = 7 or x = -24 which root can be disregarded

So x = 7 and CDE is a 7-24-25 Right Triangle with < BCE = 74 from Problem 1571
Therefore <C = 74 - 45 = 29

Sumith Peiris
Moratuwa
Sri Lanka