Ask question

Ask question

All categories

2 years ago

13

For ΔABC, ∠A = 3x - 8, ∠B = 5x - 6, and ∠C = 4x + 2. If ΔABC undergoes a dilation by a scale factor of 1 2 to create ΔA'B'C' wit

h ∠A' = 2x + 8, ∠B' = 90 - x, and ∠C' = 5x - 14, which confirms that ΔABC∼ΔA'B'C by the AA criterion?

2 answers:

quester [9]2 years ago

6 0

We all know that a triangle has 180 degrees.
3x-8+5x-6+4x+2=180
12x-12=180
12x=180+12
12x=192
x=16.

∠A=2(16)+8=40
∠B= 90-16=74
∠C = 5(16) -14= 66

-Dominant- [34]2 years ago

4 0

Answer:

$\triangle ABC \sim \triangle A'B'C'$

Step-by-step explanation:

We are given the following information in the question:

$\triangle ABC \sim \triangle A'B'C'$

$\angle A = 3x - 8, \angle B = 5x - 6, \angle C = 4x + 2.$

$\angle A' = 2x + 8, \angle B' = 90 - x, \angle C' = 5x - 14$

According to angle sum property of triangle the sum of all the three angles of triangle is 180.

$\triangle ABC\\\angle A + \angle B + \angle C = 180^\circ\\3x -8+5x-6+4x+2 = 180\\12x-12 = 180\\12x = 192\\x = 16$

For the two triangles to be similar by AA criterion.

$\angle A = \angle A'\\\angle B = \angle B'$

$\angle A = 3x - 8 = 40\\\angle A' = 2x + 8 = 40\\\angle B = 5x - 6 = 74\\\angle B' = 90 - x = 74$

Thus, this confirms that the triangles are congruent.

$\triangle ABC \sim \triangle A'B'C'$

You might be interested in

Evaluate the algebraic expression below for the given value.

MrRa [10]

hi

i think this will help you

4 0

2 years ago

Rewrite in simplest radical form 1 . show each step of your process.

morpeh [17]

Answer:

what is this russian math?

Step-by-step explanation:

4 0

3 years ago

WHAT IS THE REMAINDER WHEN <img src="https://tex.z-dn.net/?f=32%5E%7B37%5E%7B32%7D%20%7D" id="TexFormula1" title="32^{37^{32} }"

Feliz [49]

Recall Euler's theorem: if $\gcd(a,n) = 1$ , then

$a^{\phi(n)} \equiv 1 \pmod n$

where $\phi$ is Euler's totient function.

We have $\gcd(9,32) = 1$ - in fact, $\gcd(9,32^k)=1$ for any $k\in\Bbb N$ since $9=3^2$ and $32=2^5$ share no common divisors - as well as $\phi(9) = 6$ .

Now,

$37^{32} = (1 + 36)^{32} \\\\ ~~~~~~~~ = 1 + 36c_1 + 36^2c_2 + 36^3c_3+\cdots+36^{32}c_{32} \\\\ ~~~~~~~~ = 1 + 6 \left(6c_1 + 6^3c_2 + 6^5c_3 + \cdots + 6^{63}c_{32}\right) \\\\ \implies 32^{37^{32}} = 32^{1 + 6(\cdots)} = 32\cdot\left(32^{(\cdots)}\right)^6$

where the $c_i$ are positive integer coefficients from the binomial expansion. By Euler's theorem,

$\left(32^{(\cdots)\right)^6 \equiv 1 \pmod9$

so that

$32^{37^{32}} \equiv 32\cdot1 \equiv \boxed{5} \pmod9$

7 0

1 year ago

a regular rectangular pyramid has a base and lateral faces that are congruent equilateral triangles. it has a lateral surface ar

Margarita [4]

A pyramid is regular if its base is a regular polygon, that is a polygon with equal sides and angle measures.
(and the lateral edges of the pyramid are also equal to each other)

Thus a regular rectangular pyramid is a regular pyramid with a square base, of side length say x.

The lateral faces are equilateral triangles of side length x.

The lateral surface area is 72 cm^2, thus the area of one face is 72/4=36/2=18 cm^2.

now we need to find x. Consider the picture attached, showing one lateral face of the pyramid.

by the Pythagorean theorem:

$h= \sqrt{ x^{2} - (x/2)^{2}}= \sqrt{ x^{2}- x^{2}/4}= \sqrt{3x^2/4}= \frac{ \sqrt{3} }{2}x$

thus,

$Area_{triangle}= \frac{1}{2}\cdot base \cdot height\\\\18= \frac{1}{2}\cdot x \cdot \frac{ \sqrt{3} }{2}x\\\\ \frac{18 \cdot 4}{ \sqrt{3}}=x^2$

thus:

$x^2 =\frac{18 \cdot 4}{ \sqrt{3}}= \frac{18 \cdot 4 \cdot\ \sqrt{3} }{3}=24 \sqrt{3}$ (cm^2)

but $x^{2}$ is exactly the base area, since the base is a square of sidelength = x cm.

So, the total surface area = base area + lateral area = $24 \sqrt{3}+72$ cm^2

Answer: $24 \sqrt{3}+72$ cm^2

4 0

2 years ago

Which function is a quadratic function?

DerKrebs [107]

The correct answer is: [C]: " c(x) = –8x²<span> + 3x – 5 " .
_________________________________________________________
The quadratic function takes the form of:
_________________________________________________________
" f(x) = ax</span>² + bx + c " .
_________________________________________________________
Answer choice: [C]: " c(x) = –8x² + 3x – 5 " ;

is the only answer choice provided that takes that form of the quadratic function:

" f(x) = ax² + bx + c " :

in which: " a = -8 " ; " b = 3 " ; " c = -5 " .
____________________________________________________

5 0

2 years ago