6. Construct two triangles with equal bases, and angles at the bases respectively equal. Are the triangles equal?

AE

7. A person wishing to ascertain the length, AB, of a pond, places a pole at a convenient point, C, visible from A and B. The distance BC is measured, and a pole is set up, on a line with B and C, at D, the distance CD being made equal to BC. A pole is also placed at E, on a line with A and C, the distance CE D being made equal to AC.

E

Can you show that the length, AB, of the pond can be ascertained by measuring the distance DE?

CALCULATING HEIGHTS AND DISTANCES.

640. To verify the results obtained by calculation, the pupil should make diagrams, drawing the figures to a convenient scale.

Aw: AB:: wv: BC; i.e. 10: 120 :: 5: BC.

B

2. A post 6 feet above ground throws a shadow of 71⁄2 feet. How high is a tree whose shadow measures 60 feet?

3. Wishing to ascertain the distance between two houses, R and S, on opposite sides

of a stream, I measure a line, SV, at right angles to SR, 200 feet. At T, 90 feet from V, the perpendicular TW measures 60 feet. Required the distance SR.

VT: TW:: VS: SR

4. Beginning at B, 100 feet line, BC, is measured 1200 feet long. At D, distant from C 50 feet, the perpendicular DE is found to measure 90 feet. What is the distance from B to A, a tree on the opposite bank? How wide is the river?

VTVS-ST.

from the bank of a river, a

5. A boy, whose eye (E) is 4 feet from the ground, can just see the top (A) of a steeple when he stands back 3 feet from a fence (CG) 6 feet high. The distance from the foot of the fence to the centre of the base of the steeple is 177 feet. Find the height of the steeple, AB.

CD

=

? EH ?
=

ED: CD:: EH : AH.

When AH is found, how may you get AB?

6. Wishing to ascertain the distance AB, I measure a

7. Wishing to find the height of a tower, fi, I set up a pole, cd, 12 feet long above the ground. Another pole, ab, 4 feet above ground, is set up at such a distance that the tops of the two poles

The similar triangles aec and ahf give us the proportion

ae: ah ec: hf.

kij

What is the distance ec? ah=bi=bd+dk+ki. ki={kj. When fh is found, what must be added

M

he could just see M, N

R

the top of the building. He then sighted down from b, and marked on the ground the point R, on a line with ab. PQ was found by measurement to be 4 feet, RP 6 feet, PN 120 feet. Required, MN.

9. Wood-choppers, desiring to know the height of a tree before cutting it, sometimes make an isosceles right-angled triangle of wood or paper, and "step off" the distance on level ground from the point at which they find they can just see the top of the tree looking along

the hypotenuse of the triangle,

C

the base being parallel to the ground.

How high is the tree AB, if AC is 36 paces of

3 feet each, and the angle ACB is 45°?

A boy

10. B is a point on the bank of a stream due east of A on the other bank. walks due south of A until he reaches a point at which he finds, from his pocket compass, that he is directly southwest of B. If the distance AC measures 119 yards, how wide is the stream?

B

A