Question

I need help with this geometry question can someone please help me?

Answer 1

triangle LMN is an isosceles triangle (option A),

obtuse (option F)

Step-by-step explanation:

To determine the triangle which have those vertices, we need to find the distance between the points:

Given vertices:

L(-2, 4), M(3,2), N(1, -3)

Using the formula for distance:

`dis\tan ce\text{ = }\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}`
`\begin{gathered} \text{distance LM = }\sqrt[]{(2-4)^2+(3-(-2))^2} \\ \text{distance LM = }\sqrt[]{(-2)^2+(5)^2}\text{ = }\sqrt[]{29} \\ \\ \text{distance MN = }\sqrt[]{(-3-2)^2+(1-3)^2} \\ \text{distance MN = }\sqrt[]{(-5)^2+(-2)^2}\text{ = }\sqrt[]{29} \end{gathered}`
`\begin{gathered} dis\tan ce\text{ LN = }\sqrt[]{(-3-4)^2+(1-(-2))^2} \\ dis\tan ce\text{ LN = }\sqrt[]{(-7)^2+(3)^2}\text{ = }\sqrt[]{58} \end{gathered}`

From the result of the distance between the points, we find two of the distance are the same.

A triangle with two equal side and two equal angles is an isosceles triangle.

Hence, we can say triangle LMN is an isosceles triangle (option A)

A triangle can be classified as acute, obtuse or right angled.

Acute: if all 3 angles are less than 90 degrees

Obtuse: if one of the three angles is greater than 90 degrees

The two angles of an isosceles triangle are always equal and are acute traingles.

We need to check if the third will always be acute.

Isosceles triangle can either be acute or obtuse.

There is a formula for determining acute angle:

`\begin{gathered} a^2+b^2>c^2,\text{ }a^2+c^2>b^2,\text{ }b^2+c^2>a^2 \\ \text{if a = }\sqrt[]{29},\text{ b = }\sqrt[]{29},\text{ c = }\sqrt[]{58} \\ a^2+b^2>c^2\text{ : (}\sqrt[]{29})^2\text{ + }\sqrt[]{29})^2\text{ > }\sqrt[]{58})^2 \\ 58\text{ > 58 (wrong)} \\ a^2+c^2>b^2\text{ : (}\sqrt[]{29})^2\text{ + }\sqrt[]{58})^2\text{ > }\sqrt[]{29})^2 \\ 87\text{ > 29 (right)} \\ b^2+c^2>a^2\text{ : (}\sqrt[]{29})^2\text{ + }\sqrt[]{58})^2\text{ > }\sqrt[]{29})^2 \\ 87\text{ > 29 (right)} \end{gathered}`

Since one of the rule we tested is wrong, it not an acute angle.

It is an obtuse angle