jugoslav

Find the angle between the base and the altitude that is drawn to one of the legs.

That angle is 50º, but it's not 100/2 = 50, it's rather 40+10 = 50 

∆ ABC has angle bisectors AK and BM such that AK=BM=AB. Find all angle measures of ∆ ABC, in degrees.

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Triangle ABC is an isosceles triangle.

∠A = 72º           ∠B = 72º           ∠C = 36º 

Let ABCD be a rectangle with sides AB = 3 and BC = 5.  The rectangle is folded so that C ends up on AB with AC = 2 and BC = 1.  Find the area of quadrilateral PCMN.

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Information of folded rectangle: 

AB =3      AC = 2        BC = 1         CM + BM = 5

CM² - (5 - CM)² = 1           CM = 2.6        BM = 2.4

∠ACP ≅ ∠BMC   ==>         β = tan^-1(1/2.4)

DN = 2           AN = 3

[PCMN] = 6 ¹/₁₅

A triangle has sides of 6, 8, and 10 inches.  What is the distance between the incenter and circumcenter of the triangle?

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Inradius:  r = 2     

Circumradius:  R = 5

The distance between the incenter and circumcenter is √5     

A rectangle contains two circles of radius 24 and a circle of radius 16, as shown below.  Each circle is tangent to the other two circles.  Find the area of the rectangle.

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Rectangle length      L = 24 * 4 = 96

Rectangle width       W = sqrt(40² - 24²) + 24 + 16 = 72

Rectangle area        A = 96 * 72 = 6912 

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There're 3 different answers to this question!?! I understand that there could be more than one solution, but there must be only one correct answer!!! 

An Isosceles Triangle ABC, CA = CB. DEA is congruent to BEA, and DEA = BEA = 90 degrees. Circles M, N, O have radius 1 and are inscribed in ABC. What is the area of ABC?

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Triangles DEA and BEA are 3-4-5 right-angled triangles!!!

Knowing that ​it's very easy to calculate the area of ΔABC.

[ABC] ≈ 21.429 square units  

The radius of the smaller circle in the figure below is 6 units and the area of the overlapping rectangle is 243 sq units.  How many units is the radius of the larger circle?

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A = 243         r = 6         R = ?

OQ = R + 6           PQ = R - 6           OP = 243 / R

(R + 6)² - (R - 6)² = (243 / R)²

R = 13.5 units 

An equilateral triangle is inscribed within a circle, and a smaller circle is inscribed between the two shapes, tangent to both, such that it is the largest possible circle that can be inscribed.  Find the ratio of the area of the larger circle to the smaller circle.

The ratio is 16 : 1

Let ABCD be a square with a side length of 1. Points E and F lie on line BC and line CD, respectively, in such a way that angle EAF=45 degrees. If [CEF] = 1/8, what is the value of [AEF]?

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CF = 0.305            CE = 0.82          [CEF] = 1/8

[AEF] = 7/16  

MQ / 13 = (14 - MQ) / 21           MQ = ⁹¹/₁₇           NQ = 14 - MQ

AQ = sqrt(MQ² + 13²)              AQ = 14.05894659

CQ = sqrt(NQ² + 21²)              CQ = 22.71060603

AC = AQ + CQ                         AC = 36.76955262

AB = sqrt(AC² / 2)               AB = 26 units