Solutions and Notes to Supplementary Problems 1st. Edition by Mark Saul | Holooly

Solutions and Notes to Supplementary Problems

89 SOLVED PROBLEMS

Question: P.417b

Given a quadrilateral with sides a, b, c, d, diagonals e, f, and area S, we have 4e² f² = (a² + c² − b² − d²)² + 16S ². The angle V of the diagonals is given by tan V = 4S/a² + c² – b² – d². Deduce from this a solution of the preceding exercise. (Having fixed the position of one side, each of the ...

Verified Answer:

We use figure t417a, letting AB = a, BC = b, CD = ...

Question: P.420

The radii of the circles circumscribing (Exercise 66) the quadrilaterals determined by the bisectors of the interior (or exterior) angles of a quadrilat-eral are in the ratio a+c-b-d/a+c+b+d, where a, b, c, d are the sides of the given quadrilateral, taken in their natural order. ...

Verified Answer:

Suppose the given quadrilateral is MNPQ, and suppo...

Question: P.419

Among all the closed curves of same length, the circle is the one whose interior has the largest area. (Consider the ratio S/p² for a polygon inscribed in a circle and a polygon inscribed in a curve of the same length, the number of sides being the same in the two cases, and let the number of sides ...

Verified Answer:

We consider a circle with circumference C, and a c...

Question: P.418b

Among all polygons with the same number of sides and the same perimeter, the largest is the regular polygon. (Assuming that a polygon of maximum area exists, we can use the preceding exercises and Exercise 331 to show that this polygon must be regular.) The result can be restated as follows: If S ...

Verified Answer:

Suppose ABCDE... is the polygon referred to, with ...

Question: P.418

Construct a cyclic quadrilateral knowing its sides. ...

Verified Answer:

Suppose (fig. t418) ABCD is the required quadrilat...

Question: P.416

Construct a triangle knowing a side, the perimeter, and the area (construct the figure formed by the inscribed circle and an escribed circle). Among all triangles with a given side and given area, which has the smallest perimeter? Among all triangles with a given side and given perimeter, which has ...

Verified Answer:

The solution to this problem is only slightly diff...

Question: P.415

Construct a triangle knowing an angle, the perimeter, and the area (Exercises 90b, 299). Among all triangles with a given angle and given perimeter, which one has largest area? ...

Verified Answer:

Suppose we know the measure of angle

\hat{{...

Question: P.414

Another solution to Exercise 329: to draw through a given point inside an angle a secant which forms, with the sides of the angle, a triangle with given area. Construct first the parallelogram with a vertex at the given point, and two sides on the sides of the angle. This parallelogram cuts from ...

Verified Answer:

Suppose (fig. t414) we must draw secant MON throug...

Question: P.413b

If four circles are inscribed in the same angle, or in the corresponding vertical angle, and they are also tangent to a fifth circle, then their radii r1, r2, r3, r4 form a proportion. (One observes that these circles can be arranged in pairs which correspond to each other in the same inversion.) ...

Verified Answer:

Lemma. If two tangent circles are inverted around ...

Question: P.413

We are given two parallel lines, and a common perpendicular which intersects them in A, B. Points C, D are taken on these lines so that trape-zoid ABCD has an area equal to that of a given square. Let H be the projection on CD of the midpoint of AB. Find the locus of H. (One must distinguish two ...

Verified Answer:

One key to this problem is the observation that be...

Loading...