Question

question(+10) how do we know if a figure is an isosceles triangle? you have to check whether it has two congruent sides and angles. part i show all the necessary steps! for problems 1 - 6, find the distance between each pair of points. round to nearest tenth when necessary. 1. (0,0) and (-3,4) 2. (1,2) and (6,14) 3. (8,11) and (15,35) 4. (3,8) and (-5,-7) 5. (-2,-3) and (-1,-4) 6. (-3,5) and (3,5) 7. the points a(0,0), b(3,4), and c(-1,1) form a triangle. is it an isosceles triangle? justify your answer.

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Solve problem 1

For points $(0,0)$ and $(-3,4)$, substitute $x_1 = 0,y_1 = 0,x_2=-3,y_2 = 4$ into the formula:

$$
LATEXBLOCK0
$$

Step3: Solve problem 2

For points $(1,2)$ and $(6,14)$, substitute $x_1 = 1,y_1 = 2,x_2=6,y_2 = 14$ into the formula:

$$
LATEXBLOCK1
$$

Step4: Solve problem 3

For points $(8,11)$ and $(15,35)$, substitute $x_1 = 8,y_1 = 11,x_2=15,y_2 = 35$ into the formula:

$$
LATEXBLOCK2
$$

Step5: Solve problem 4

For points $(3,8)$ and $(-5,-7)$, substitute $x_1 = 3,y_1 = 8,x_2=-5,y_2 = -7$ into the formula:

$$
LATEXBLOCK3
$$

Step6: Solve problem 5

For points $(-2,-3)$ and $(-1,-4)$, substitute $x_1=-2,y_1 = -3,x_2=-1,y_2 = -4$ into the formula:

$$
LATEXBLOCK4
$$

Step7: Solve problem 6

For points $(-3,5)$ and $(3,5)$, substitute $x_1=-3,y_1 = 5,x_2=3,y_2 = 5$ into the formula:

$$
LATEXBLOCK5
$$

Step8: Solve problem 7

Find the lengths of the sides of the triangle.
For side $AB$ with $A(0,0)$ and $B(3,4)$:

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LATEXBLOCK6
$$

For side $AC$ with $A(0,0)$ and $C(-1,1)$:

$$
LATEXBLOCK7
$$

For side $BC$ with $B(3,4)$ and $C(-1,1)$:

$$
LATEXBLOCK8
$$

Since $AB = BC=5$, the triangle $ABC$ is an isosceles triangle.

Answer:

1. $5$
2. $13$
3. $25$
4. $17$
5. $\approx1.4$
6. $6$
7. It is an isosceles triangle.