MATH SOLVE

4 months ago

Q:
# WILL GIVE BRAINLIEST! (See attachment)A. If AB is 20 and BC is 21, based only on the appearance of the figure, what is a good estimate for the length of AC? How do you know?B. If the actual length of AC is exactly what you computed in part A above, what is the measure of angle ABC? How do you know?C. Given the information from A and B above, if AE is 10 and A.F is half of AC, what is the special name for segment EF as it relates to triangle ABC?D. Given the result from C above, if the measure of angle EFB is 43.6°, what is the measure of angle FBC? How do you know?E. If Triangle DCB is congruent to triangle ABC, what kind of shape is ABCD? How do you know?

Accepted Solution

A:

Part A:

From the appearance of the figure above, triangle A B C forms a right angle at B with A B and B C being the legs and A C being the hypothenuse.

Given that A B is 20 and B C is 21, by the pythagoras theorem,

[tex]A B^2+B C^2=A C^2 \\ \\ \Rightarrow20^2+21^2=A C^2 \\ \\ \Rightarrow400+441=A C^2 \\ \\ \Rightarrow841=A C^2 \\ \\ \Rightarrow A C=\sqrt{841}=29[/tex]

Part B:

From the appearance of the figure above, triangle A B C forms a right angle at B with A B and B C as the legs and A C as the hypothenuse.

If this is true, then the measure of angle A B C is 90 degrees.

Part C:

If A E is 10 and A F is half of A C, the special name for segment E F as it relates to triangle A B C is the midsegment.

The midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle.

Part D:

Given from part C above that line segment EF is a midsegment of triangle ABC, from the triangle midsegment theorem, line segment EF is parallel to side BC.

Thus, line BF is a transverse of parallel lines EF and BC which makes angle EFB alternate to angle FBC.

Since alternate angles are equal, given that angle EFB is 43.6°, then the measure of angle FBC is also 43.6° because they are alternate angles.

Part E:

If Triangle DCB is congruent to triangle ABC, then angle B is congruent to angle C.

Given from part B that angle B = 90 degrees, then angle C = 90 degrees. Thus, the two adjacent angles of the quadrilateral ABCD = 90 degrees. Whih shows that the shape ABCD is a rectangle.

From the appearance of the figure above, triangle A B C forms a right angle at B with A B and B C being the legs and A C being the hypothenuse.

Given that A B is 20 and B C is 21, by the pythagoras theorem,

[tex]A B^2+B C^2=A C^2 \\ \\ \Rightarrow20^2+21^2=A C^2 \\ \\ \Rightarrow400+441=A C^2 \\ \\ \Rightarrow841=A C^2 \\ \\ \Rightarrow A C=\sqrt{841}=29[/tex]

Part B:

From the appearance of the figure above, triangle A B C forms a right angle at B with A B and B C as the legs and A C as the hypothenuse.

If this is true, then the measure of angle A B C is 90 degrees.

Part C:

If A E is 10 and A F is half of A C, the special name for segment E F as it relates to triangle A B C is the midsegment.

The midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle.

Part D:

Given from part C above that line segment EF is a midsegment of triangle ABC, from the triangle midsegment theorem, line segment EF is parallel to side BC.

Thus, line BF is a transverse of parallel lines EF and BC which makes angle EFB alternate to angle FBC.

Since alternate angles are equal, given that angle EFB is 43.6°, then the measure of angle FBC is also 43.6° because they are alternate angles.

Part E:

If Triangle DCB is congruent to triangle ABC, then angle B is congruent to angle C.

Given from part B that angle B = 90 degrees, then angle C = 90 degrees. Thus, the two adjacent angles of the quadrilateral ABCD = 90 degrees. Whih shows that the shape ABCD is a rectangle.