91Ó°ÊÓ

Exercises \(23-26,\) draw the indicated triangle and \(\square\)nd its centroid and orthocenter. isosceles right triangle

The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, state an inequality involving the sum of the angles of a triangle. Find a formula for the area of a triangle in spherical geometry

Is it possible to construct a triangle with the given side lengths? If not, explain why not. $$35,120,125$$

Exercises 27 and 28, describe and correct the error in \(\square\) nding DE. Point \(\mathrm{D}\) is the centroid of \(\Delta \mathrm{ABC} .\) $$\mathrm{DE} \square \frac{2}{3} \mathrm{AE}$$ $$\mathrm{DE} \square \frac{2}{3} \mathrm{(18)}$$ $$\mathrm{DE} \square 12$$

ATTENDING TO PRECISIOn The points \(\mathrm{P}(2,1)\) \(\mathrm{Q}(4,5),\) and \(\mathrm{R}(7,4)\) are the midpoints of the sides of a triangle. Graph the three midsegments. Then show how to use your graph and the properties of midsegments to draw the original triangle. Give the coordinates of each vertex.

MODELING WITH MATHEMATICS are placing a fountain in a triangular koi pond. You want the fountain to be the same distance from each edge of the pond. Where should you place the fountain? Explain your reasoning. Use a sketch to support your answer.

Your class has fewer than 30 students. The teacher divides your class into two groups. The first group has 15 students. Use indirect reasoning to show that the second group must have fewer than 15 students.

CRITICAL THINKING Exercises \(29-32,\) complete the statement with always, sometimes, or never. Explain your reasoning. If the perpendicular bisector of one side of a triangle intersects the opposite vertex. then the triangle is _____ isosceles.

MAKING AN ARGUMENT Your friend says it is impossible for an angle bisector of a triangle to be the same line as the perpendicular bisector of the opposite side. Is your friend correct? Explain your reasoning.

CRITICAL THINKING Exercises \(29-32,\) complete the statement with always, sometimes, or never. Explain your reasoning. The perpendicular bisectors of a triangle intersect at a point that is ________ equidistant from the midpoints of the sides of the triangle.