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3 years ago

Find the value of x in the isosceles triangle shown below.

Mathematics

1 answer:

Solnce55 [7]3 years ago

8 0

Answer: B. x=10

Step-by-step explanation:

Concept

- If it is an isosceles triangle, then the altitude would also be the median of the base, meaning that it divides the base evenly.

- We are going to use the Pythagorean theorem which is a²+b²=c², where [a] and [b] are legs, and [c] is the hypotenuse.

Given

Hypotenuse (relative) = √74

Altitude (one leg) = 7

The other leg (half of the base) = x/2

Solve

a² + b² = c²

(7)² + (x/2)² = (√74)²

49 + x²/4 = 74

x²/4 = 25

x² = 100

x = 10

Hope this helps!! :)

Please let me know if you have any questions

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n is a positive integer with exactly two different divisors greater than 1, how many positive factors does n^2 have? A. 4 B. 5 C

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Answer:

C. 5

Step-by-step explanation:

Remember that for any integer $n$ , the integers $1 \text{and} n$ are both divisors (or factors) of $n$ . First, we will prove that n is a square, and then we will compute the factors of the n².

In this case, the integer n has exactly two different divisors greater than 1. It's impossible that $n=1$ , since 1 doesn't have positive factors greater than 1. Then $n>1$ , therefore $n$ itself is one of the required divisors. Denote by $a$ the other divisor greater than 1, and note that to satisfy the condition on the divisors, $a$ .

Because a divides n, there exists some integer $k$ such that $n=ak$ . We must have that $k>1$ , if not, then $k\leq 1$ , which implies that $ak=n\leq a$ , which contradicts the part above.

Now, $k>1$ and, by definition of divisibility, k divides n. Then k must be equal either to n or a, since we can't have three different divisors of this kind. If $k=n$ then $n=an$ and by cancellation, $1=a$ which is a contradiction. Therefore $k=a$ and $n=a^2$ .

We have that $n=(a^2)^2=a^4$ . We can write n as $n=a^3a=a^2a^2=a^4 \cdot 1$ . From the first equation, a divides n and a³ divides n. From the second equation, a² divides, and from the last one, 1 divides n and a⁴=n divides n. Thus n has exactly 5 positive factors.