An angle bisector of a triangle divides the opposite side of the triangle into segments 5 cm and 3 cm long. A second side of the
triangle is 7.6 cm long. Find the longest and shortest possible lengths of the third side of the triangle. Round answers to the nearest tenth of a centimeter.
a. 12.7 cm, 4.6 cm
b. 15 cm, 4.6 cm
c. 38 cm, 2 cm
d. 38 cm, 12.7 cm
a. 12.7 cm, 4.6 cm
b. 15 cm, 4.6 cm
c. 38 cm, 2 cm
d. 38 cm, 12.7 cm
1 answer:
There is a little-known theorem to solve this problem.
The theorem says that
In a triangle, the angle bisector cuts the opposite side into two segments in the ratio of the respective sides lengths.
See the attached triangles for cases 1 and 2. Let x be the length of the third side.
Case 1:
Segment 5cm is adjacent to the 7.6cm side, then
x/7.6=3/5 => x=7.6*3/5=4.56 cm
Case 2:
Segment 3cm is adjacent to the 7.6 cm side, then
x/7.6=5/3 => x=7.6*5/3=12.67 cm
The theorem can be proved by considering the sine rule on the adjacent triangles ADC and BDC with the common side CD and equal angles ACD and DCB.
The theorem says that
In a triangle, the angle bisector cuts the opposite side into two segments in the ratio of the respective sides lengths.
See the attached triangles for cases 1 and 2. Let x be the length of the third side.
Case 1:
Segment 5cm is adjacent to the 7.6cm side, then
x/7.6=3/5 => x=7.6*3/5=4.56 cm
Case 2:
Segment 3cm is adjacent to the 7.6 cm side, then
x/7.6=5/3 => x=7.6*5/3=12.67 cm
The theorem can be proved by considering the sine rule on the adjacent triangles ADC and BDC with the common side CD and equal angles ACD and DCB.
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Answer:
3 metres
Step-by-step explanation:
If we draw this out, we'll see that there are actually two similar right triangles (see attachment), which means that we can set up a proportion.
The height of the lookout tower corresponds to the height of the wooden column, while the shadow of the lookout tower corresponds to the shadow of the wooden column. We can then write:
(height of lookout tower) / (shadow of tower) = (height of column) / (shadow of column)
16 / 12 = 4 / x , where x is the shadow / unknown we want to find
Cross-multiply:
16x = 48
x = 3
The answer is thus 3 metres.
Alright, lets get started.
Maria hiked the 5 mile Mountain Trail a number of times and she hiked the 10 mile canal trail several times.
x times she hiked 5 mile mountain.
So, number of mils she hiked for 5 miles will be = 5 x
Similarly,
y times she hiked 10 mile canal,
So, number of miles she hiked for canal = 10 y
So, total number of miles she hiked =
But as per question given, she had a total of 90 miles
It means
Dividing this equation with 5
So, the equation is : Answer
Hope it will help :)