Question

1.Find the distance between the two labeled points. (picture 1)

A)2
B)5
C)50
D)25

2. What theorem can be used to prove that the two triangles are congruent?
A) HL
B) LA
C) LL
D) SSS

Answer 1

Part (1):

The distance between two points (x₁,y₁),(x₂,y₂) = d

`d = \sqrt{ (x2-x1)^(2) + (y2-y1)^(2) }`

The coordinates of the two labeled points are:

(2,3) which are in the first quadrant

(-3,-2) which are in the third quadrant

∴
`d = \sqrt{ (-3-2)^(2) + (-2-3)^(2) }` = 5√2

So, the distance between the two labeled points = 5√2

The correct answer is option B)5√2

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Part (2):

Both of ΔABC and ΔDEF are right triangles

We can conclude the following:

1) ∠B = ∠E = 90°

2) AC = DF ⇒⇒⇒ Hypotenuse

3) BC = EF ⇒⇒⇒ Leg

So, the theorem that can be used to prove that the two triangles are congruent is HL

The correct answer is option A) HL

Answer 2

Answers:
1.√50
2. A)HL.

Step-by-step explanation:
The first problem is solved using the distance formula.
Distance between these two points =
`\sqrt{( x_(2)- x_(1))^2 + ( y_(2) - y_(1) )^2 }`
The Coordinates of two points are (-3, -2) and (2, 3), therefore:
x₁= -3,
x₂ = 2 ,
y₁ = -2 ,
y₂ = 3.

Plug in values to get:

`√(( 2- -3)^2 + ( 3) - -2 )^2 }` = √50.

The second problem has a figure that is a Right triangle and HL is used when a right triangle as a the Hypotenuse and one leg that is congruent and that is designated by the corresponding tick marks on the image.