Chapter 10.3, Problem 4CE

To determine

in the given triangle JAM:

1. Is JM an Altitude of triangle JAM?
2. What is another altitude?
3. At what point all three altitudes meet?
4. Where all the three perpendicular bisectors meet?
5. Does theorem 10-2 supports (d)?

Explanation of Solution

Given:

JAM is a right-angled triangle.

Calculation:

a.

Line JM is perpendicular to the side MA.

Therefore, the line JM is an altitude of the triangle JAM from the vertex J.

b.

By observing the given figure:

Line AM is also an altitude of this triangle from the vertex A.

c.

From the point M draw two arcs on the line JA with equal lengths.

From X and Y draw other two arcs and mark their intersection as a point Z and join MZ.

MXZ and JA intersect at point D.

MD is an altitude of triangle JAM that contains the vertex M.

  

d.

The perpendicular bisectors of a right-angled triangle meet at a point on the hypotenuse of this triangle as shown below:

  

e.

In the given figure,

  $\begin{array}{l}\overline{XJ}=\overline{XA},\\ \overline{XJ}=\overline{XM},\\ \Rightarrow \overline{XA}=\overline{XM}=\overline{XJ}.\end{array}$

X is equidistant from the vertices M, J, A. Hence, the theorem 10-2 supports (d).