In a right triangle, the square of the hypotenuse is equal to twicethe product of the legs. The measure?

Question

in degrees of the smallest angle in the triangle is

a)10 b)20 c)30 d)40 e)45

Please Show your work.

9)In the base 10 number system, the number 526 5 x 10^2+ 2x10+6. In the land of Oz, numbers are written in base r. Lloyd purchases a car for 440 unites. He gives the salesman 1000 units and receives 340 unites in change. What is the base r?

a) 4 b) 5 c)6 d)7 e)8

Please show your work.

Thanks

Answer 1

For a right triangle, we know:
a² + b² = h²
For this particular triangle, we are told that:
h² = 2ab
So solve...
2ab = a² + b²
a² - 2ab + b² = 0
(a - b)² = 0
a - b = 0
a = b
The length of the legs are equal; therefore, the opposite angles must be equal. Since we know that one angle is 90°,
90° + α + β = 180°
α = β
2α = 90°
α = 45°
The answer is e, 45°. The triangle is a right isoceles triangle with angles of 45°, 45°, and 90°.


Second problem:
1000
- 440
-------
=340
Since we are in a different base system, we have to write an equation of powers of r:
r³ - (4r² + 4r) = 3r² + 4r
r³ - 4r² - 4r = 3r² + 4r
r³ - 7r² - 8r = 0
r(r - 8)(r + 1) = 0
r = 8, 0, -1
Since a base cannot be zero or negative, the answer must be e. r = 8.

Answer 2

First problem:

c^2 = 2ab
a^2 + b^2 = 2ab
a^2 - 2ab + b^2 = 0
(a - b)^2 = 0
a = b

Since you said it's a right triangle, and the other sides are equal, it's isosceles, so the angles are 45 degrees.

Second question)
essentially, we're saying that in base R 440 + 340 = 1000.
In base 10, the sum is 780. The only base that causes that to convert to 1000 is 8, since the 2 4s, when added together would form 10, forcing a carry to add to the 7, making 10 again.
Solution: base 8 (E)

Answer 3

1.
a^2 + b^2 = c^2
2ab = c^2

a^2 + b^2 = 2ab
a^2 - 2ab + b^2 = 0
(a - b)^2 = 0
a - b = 0
a = b

If the 2 legs are congruent, it becomes isosceles. One of the angles is 90, and the other 2 are equal.
(180 - 90) / 2 = 45

The other two angles are 45. E.